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Networks constructed out of resistors, inductors, capacitors and transformersform a compelling subclass of simple models. Models constructed out of thesebasic elements are frequently used to explain phenomena in large-scaleapplications, from inter-area oscillations in power systems, to the transientbehaviour of optimisation algorithms. Furthermore they capture the dynamics ofthe most commonly applied controllers, including the PID controller. In thispaper we show that the inherent structure in these networks can be used tosimplify, or even solve analytically, a range of simple optimal controlproblems. We illustrate these results by designing and synthesising simple,scalable, and globally optimal control laws for solving constrained leastsquares problems, regulating electrical power systems with stochastic renewablesources, studying the robustness properties of consensus algorithms, andanalysing heating networks.

Passive and reciprocal networks: From simple models to simple optimal controllers